Video Speed Changes when rendering
Blender may be buffering the source video differently at render? Try transcoding the source files to another codec first, as it looks like a screen capture and they are notorious for having strange FPS (frames per second) values.

I'm afraid that Blender's video function is handled by a limited version of ffmpeg that may not recognize your file correctly.

Otherwise check your output settings, there might be something odd about those codec settings too?